By enabling the Google calendar integration start and due dates on cards that's assigned to you will show up in your Google Calendar. Once you have enabled it for your organization all the members of the organization can choose to connect their account with their Google calendar.
To enable the Google Calendar integration press your avatar in top right corner, go to the App Store and choose Integrations.
When enabled for your organization all members will have the option to connect their Google calendar by clicking the avatar and Edit profile See picture below.
Remember that cards assigned to you with start and due dates created before enabling the integration will not appear in your Google calendar. Only cards created, assigned to you with start and due date, after enabling the integration will be visible in your Google calendar.